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Is there a nice way to characterize the singular matrices whose principal minors up to some size $r\times r$ are always non-vanishing?

I was wondering whether there is some theorem regarding this, since I often see such matrices when discretising elliptic boundary value problems with the finite element method. For instance the stiffness matrix when discretising the Laplacian is a symmetric positive semi-definite matrix of rank $n-1$, and any of its principal submatrices of size $\leq n-1$ are positive definite.

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  • $\begingroup$ Matrices with non-vanishing principal minors up to size $r \times r$ are rank-deficient. In the context of elliptic boundary value problems, such matrices are symmetric positive semi-definite (SPSD) with one zero eigenvalue. Also Sylvester's criterion may be useful in this context. $\endgroup$
    – rumathe
    Mar 19 at 20:59
  • $\begingroup$ @rumathe You can have more zero eigenvalues depending on how many boundary conditions you specify. I agree that in the usual case of a Poisson problem there is one zero eigenvalue corresponding to the constant eigenvector. Once a Dirichlet condition is specified the problen becomes well-posed and the matrix symmetric positive definite. This happens by cutting out the rows and columns corresponding to the Dirichlet data. Sylvester's criterion only says that the minors of a symmetric oositive semi-definite matrix are non-negative. What I want is quite different. $\endgroup$
    – lightxbulb
    Mar 19 at 22:41

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